Tame and strongly \'etale cohomology of curves

TL;DR

This paper investigates the tame cohomology of curves over perfect fields in characteristic p, establishing cohomological purity and Poincaré duality for the tame cohomology groups with p-torsion coefficients, which do not hold for full étale cohomology.

Contribution

It proves cohomological purity and Poincaré duality for tame cohomology of curves over perfect fields, highlighting differences from full étale cohomology.

Findings

Tame cohomology satisfies cohomological purity.

Poincaré duality holds for tame cohomology with p-torsion.

Purity does not hold in full étale cohomology.

Abstract

For a curve $C$ over a perfect field $k$ of characteristic $p > 0$ we study the tame cohomology of $X = \textit{Spa}(C,k)$ introduced in arXiv:1801.04776. We prove that the tame cohomology groups of $X$ with $p$-torsion coefficients satisfy cohomological purity (which is not true in full generality for the \'etale cohomology). Using purity we show Poincar\'e duality for the tame cohomology of $X$ with $p$-torsion coefficients.